# Likelihood function for a multiperiod probit with autoregressive latent variable

I'd like to evaluate the likelihood function of a multperiod ordered probit model with an autoregressive random component, but i am having trouble arriving at the likelihood function. As an example (here: Eichengreen (1985) on the choice of bank rates $BR_{t}$ and change in bank rates $\Delta BR_{t}$). There are three observed changes -50, 0, 100. Their model is: $$Prob[\Delta{BR_{t}}=-50|J_{t}]= Prob[BR^*_{t}<BR_{t-1}-\alpha_{l}]$$ $$Prob[\Delta{BR_{t}}= 0|J_{t}]= Prob[BR_{t-1}-\alpha_{l}<BR^*_{t}<BR_{t-1}+\alpha_{U}]$$ $$Prob[\Delta{BR_{t}}= 100|J_{t}]= Prob[BR^*_{t}>BR_{t-1}+\alpha_{U}]$$ with $\Delta BR^*_{t}$ being the latent variable, $J_{t}$ containing current and lagged values of exogenous variables $x_{t}$ and the complete history of $BR_{s}$, s=1,...,t-1. For example, $J_{t}=(X_{t}, BR_{t-1},X_{t-1},...,BR_{1},X_{1})$ The underlying latent variable regression is: $$\Delta{BR^*_{t}}=X^'_{t}ß+\epsilon_{t}$$ with $\epsilon_{t}$ ~ $Niid (0,\sigma^2)$. Written in levels (solving backward): $$BR^*_{t}=BR^{*}_{0}+\sum^{t}_{i=1}X_{i}\beta+\sum^{t}_{i=1}\epsilon_{i}$$ Replacing the latent variable in the model by the solved difference equation and $\gamma_{t}=\sum^{t}_{i=1}\epsilon_{i}$ the probabilities are: $$Prob[\Delta{BR_{t}}=-50|J_{t}]= Prob[\gamma_{t}<BR_{t-1}-\alpha_{l}-BR^{*}_{0}- \sum^{t}_{i=1}X_{i}\beta]$$ $$Prob[\Delta{BR_{t}}= 0|J_{t}]=$$ $$Prob[BR_{t-1}-\alpha_{l}-BR^{*}_{0}-\sum^{t}_{i=1}X_{i}\beta<\gamma_{t}<BR_{t-1}+\alpha_{u}-BR^{*}_{0}-\sum^{t}_{i=1}X_{i}\beta]$$ $$Prob[\Delta{BR_{t}}= 100|J_{t}]= Prob[\gamma_{t}>BR_{t-1}+\alpha_{U}-BR^{*}_{0}-\sum^{t}_{i=1}X_{i}\beta|J_{t}]$$ To evaluate the probabilities i need the conditional distribution of $\gamma|J_{t}$. $\gamma$ follows a random walk $\gamma_{t}= \sum^{t}_{i=1}\epsilon_{i}=\gamma_{t-1}+\epsilon_{t}$ with $\epsilon_{t}$. The Markov structure of $\gamma_{t}$ allows to write the conditional distribution of $\gamma_{t}|J_{t}$ for the bounds on $\gamma_{t}$ by the realisation of $\Delta BR_{t}$, e.g. $$l_{t}= BR_{t-1}-\alpha_{l}-BR^{*}_{0}-\sum^{t}_{i=1}X_{i}\beta$$ and $$U_{t}= R_{t-1}+\alpha_{u}-BR^{*}_{0}-\sum^{t}_{i=1}X_{i}\beta$$ by the convolution as: $$h(\gamma_{t}|J_{t})=\frac{1}{\sigma}\int^{U_{t-1}}_{l_{t}}\phi(\frac{\gamma_{t}-\gamma_{t-1}}{\sigma})g(\gamma_{t-1}|J_{t})d\gamma_{t-1}$$ with $g(\gamma_{t-1}|J_{t})$ as the conditional distribution of $\gamma_{t-1}$ given $J_{t}$. In order to determine the likelihood function i need to find $h(\gamma_{t+1}|J_{t+1})$. To find the latter $g(\gamma_{t}|J_{t+1})$ is required. The authors suggest the following: $$g(\gamma_{t}|J_{t+1})=[H(U_{t}|J_{t})-H(l_{t}|J_{t}]^{-1} h(\gamma_{t}|J_{t})$$ for $l_{t}<\gamma_{t}<U_{t}$ and H(°|°) as the cdf of the density h(°|°).

I can follow the argumentation, but i am having trouble with the last step. Can anyone give me hint what's the rational behind 'updating' $h(\gamma_{t+1}|J_{t+1})$ by using the cdfs?

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